Found problems: 85335
In a country with $2015$ cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which $k$ flights belong to the same airline, what is the maximum value of $k$?
The sides of an equilateral triangle with sides of length $n$ have been divided into equal parts, each of length $1$, and lines have been drawn through the points of division parallel to the sides of the triangle, thus dividing the large triangle into many small triangles. Nils has a pile of rhombic tiles, each of side $1$ and angles $60^\circ$ and $120^\circ$, and wants to tile most of the triangle using these, so that each tile covers two small triangles with no overlap. In the picture, three tiles are placed somewhat arbitrarily as an illustration. How many tiles can Nils fit inside the triangle?
[asy]
/* original code by fedja: https://artofproblemsolving.com/community/c68h207503p1220868
modified by Klaus-Anton: https://artofproblemsolving.com/community/c2083h3267391_draw_me_a_grid_of_regular_triangles
*/
size(5cm);
int n=6;
pair A=(1,0), B=dir(60);
path P=A--B--(0,0)--cycle;
path Pp=A--shift(A)*B--B--cycle;
/*
label("$A$",A,S);
label("$B$",B,dir(120));
label("$(0,0)$",(0,0),dir(210));
fill(shift(2*A-1+2*B-1)*P,yellow+white);
fill(shift(2*A-1+2*B-0)*P,yellow+white);
fill(shift(2*A-1+2*B+1)*P,yellow+white);
fill(shift(2*A-1+2*B+2)*P,yellow+white);
fill(shift(1*A-1+1*B)*P,blue+white);
fill(shift(2*A-1+1*B)*P,blue+white);
fill(shift(3*A-1+1*B)*P,blue+white);
fill(shift(4*A-1+1*B)*P,blue+white);
fill(shift(5*A-1+1*B)*P,blue+white);
fill(shift(0*A+0*B)*P,green+white);
fill(shift(0*A+1+0*B)*P,green+white);
fill(shift(0*A+2+0*B)*P,green+white);
fill(shift(0*A+3+0*B)*P,green+white);
fill(shift(0*A+4+0*B)*P,green+white);
fill(shift(0*A+5+0*B)*P,green+white);
fill(shift(2*A-1+3*B-1)*P,magenta+white);
fill(shift(3*A-1+3*B-1)*P,magenta+white);
fill(shift(4*A-1+3*B-1)*P,magenta+white);
fill(shift(5*A+5*B-5)*P,heavyred+white);
fill(shift(4*A+4*B-4)*P,palered+white);
fill(shift(4*A+4*B-3)*P,palered+white);
fill(shift(0*A+0*B)*Pp,gray);
fill(shift(0*A+1+0*B)*Pp,gray);
fill(shift(0*A+2+0*B)*Pp,gray);
fill(shift(0*A+3+0*B)*Pp,gray);
fill(shift(0*A+4+0*B)*Pp,gray);
fill(shift(1*A+1*B-1)*Pp,lightgray);
fill(shift(1*A+1*B-0)*Pp,lightgray);
fill(shift(1*A+1*B+1)*Pp,lightgray);
fill(shift(1*A+1*B+2)*Pp,lightgray);
fill(shift(2*A+2*B-2)*Pp,red);
fill(shift(2*A+2*B-1)*Pp,red);
fill(shift(2*A+2*B-0)*Pp,red);
fill(shift(3*A+3*B-2)*Pp,blue);
fill(shift(3*A+3*B-3)*Pp,blue);
fill(shift(4*A+4*B-4)*Pp,cyan);
fill(shift(0*A+1+0*B)*Pp,gray);
fill(shift(0*A+2+0*B)*Pp,gray);
fill(shift(0*A+3+0*B)*Pp,gray);
fill(shift(0*A+4+0*B)*Pp,gray);
*/
fill(Pp, rgb(244, 215, 158));
fill(shift(dir(60))*P, rgb(244, 215, 158));
fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*Pp, rgb(244, 215, 158));
fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*P, rgb(244, 215, 158));
fill(shift(-.5,(-sqrt(3)/2))*shift(4*dir(60))*Pp, rgb(244, 215, 158));
fill(shift(.5,(-sqrt(3)/2))*shift(4*dir(60))*P, rgb(244, 215, 158));
for(int i=0;i<n;++i){
for(int j;j<n-i;++j)
{draw(shift(i*A+j*B)*P);}}
shipout(bbox(2mm,Fill(white)));
[/asy]
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its [i]colorfulness [/i]as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.
[i]Proposed by Pavle Martinović[/i]
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions
$$a \mid b + s(a)$$
$$b \mid a + s(b)$$
[i]Proposed by Victor Domínguez[/i]
Vertex $ E$ of equilateral $ \triangle{ABE}$ is in the interior of unit square $ ABCD$. Let $ R$ be the region consisting of all points inside $ ABCD$ and outside $ \triangle{ABE}$ whose distance from $ \overline{AD}$ is between $ \frac{1}{3}$ and $ \frac{2}{3}$. What is the area of $ R$?
$ \textbf{(A)}\ \frac{12\minus{}5\sqrt3}{72} \qquad
\textbf{(B)}\ \frac{12\minus{}5\sqrt3}{36} \qquad
\textbf{(C)}\ \frac{\sqrt3}{18} \qquad
\textbf{(D)}\ \frac{3\minus{}\sqrt3}{9} \qquad
\textbf{(E)}\ \frac{\sqrt3}{12}$
In each box of a $9\times 9$ grid we write a positive integer such that, between any $2$ boxes on the same row or column that have the same number $n$ written, there's at least $n$ boxes between them. What is the minimum sum possible for the numbers on the grid?
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.
Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of
\[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]
Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose
\[a^2+b^2+c^2+d^2=ab+bc+cd+da,\]
and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.
Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.
Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$
(i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$.
(ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.
Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$.
Given is a circle $\omega$ and a line $\ell$ tangent to $\omega$ at $Y$. Point $X$ lies on $\ell$ to the left of $Y$. The tangent to $\omega$, perpendicular to $\ell$ meets $\ell$ at $A$ and touches $\omega$ at $D$. Let $B$ a point on $\ell$, to the right of $Y$, such that $AX=BY$. The tangent from $B$ to $\omega$ touches the circle at $C$. Prove that $\angle XDA= \angle YDC$.
Note: This is not the official wording (it was just a diagram without any description).
The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD \equal{} P$ and $ AC\cap B'D \equal{} Q$ then prove that $ PQ \perp AC$
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
The tangents to the circumcircle of triangle $ABC$ at $A$ and $B$ meet at point $D$. The circle passing through the projections of $D$ to $BC, CA, AB$, meet $AB$ for the second time at point $C'$. Points $A', B'$ are defined similarly. Prove that $AA', BB', CC'$ concur.
Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes.
$\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$.
$\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$.
$\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$.
Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.
Find all possible non-negative integer solution ($x,$ $y$) of the following equation-
$$x!+2^y=z!$$
Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.
Mike leaves home and drives slowly east through city traffic. When he reaches the highway he drives east more rapidly until he reaches the shopping mall where he stops. He shops at the mall for an hour. Mike returns home by the same route as he came, driving west rapidly along the highway and then slowly through city traffic. Each graph shows the distance from home on the vertical axis versus the time elapsed since leaving home on the horizontal axis. Which graph is the best representation of Mike's trip?
[asy]
import graph;
unitsize(12);
real a(real x) {return ((x-15)^2)/2;}
real b(real x) {return ((x-25)^2)/2;}
real c(real x) {return ((x-30)^2 * (x-40)^2) * 8/625;}
real d(real x) {return ((x-15)^2)*8/25-15;}
real e(real x) {return ((x-25)^2)*8/25-15;}
draw((0,9)--(0,0)--(11,0));
draw((15,9)--(15,0)--(26,0));
draw((30,9)--(30,0)--(41,0));
draw((0,-6)--(0,-15)--(11,-15));
draw((15,-6)--(15,-15)--(26,-15));
draw((0,0)--(3,8)--(7,8)--(10,0));
draw(graph(a,15,17));
draw((17,2)--(18,8)--(22,8)--(23,2));
draw(graph(b,23,25));
draw(graph(c,30,40));
draw((0,-15)--(5,-7)--(10,-15));
draw(graph(d,15,20));
draw(graph(e,20,25));
for (int k=0; k<3; ++k)
{
label("d",(15*k-1,8),N); label("i",(15*k-1,7),N); label("s",(15*k-1,6),N); label("t",(15*k-1,5),N); label("a",(15*k-1,4),N); label("n",(15*k-1,3),N); label("c",(15*k-1,2),N); label("e",(15*k-1,1),N);
label("time",(15*k+8,0),S);
}
for (int k=0; k<2; ++k)
{
label("d",(15*k-1,8-15),N); label("i",(15*k-1,7-15),N); label("s",(15*k-1,6-15),N); label("t",(15*k-1,5-15),N); label("a",(15*k-1,4-15),N); label("n",(15*k-1,3-15),N); label("c",(15*k-1,2-15),N); label("e",(15*k-1,1-15),N);
label("time",(15*k+8,0-15),S);
}
label("(A)",(5,9),N); label("(B)",(20,9),N); label("(C)",(35,9),N); label("(D)",(5,-6),N); label("(E)",(20,-6),N);
[/asy]
Let $N_{n}$ denote the number of ordered $n$-tuples of positive integers $(a_{1},a_{2},\ldots,a_{n})$ such that \[1/a_{1}+1/a_{2}+\ldots+1/a_{n}=1.\] Determine whether $N_{10}$ is even or odd.
Let $ a$ and $ b$ be the roots of the equation $ x^2 \minus{} mx \plus{} 2 \equal{} 0$. Suppose that $ a \plus{} (1/b)$ and $ b \plus{} (1/a)$ are the roots of the equation $ x^2 \minus{} px \plus{} q \equal{} 0$. What is $ q$?
$ \textbf{(A) } \frac 52 \qquad \textbf{(B) } \frac 72 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } \frac 92 \qquad \textbf{(E) } 8$
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one side along a side of $T.$ Then inscribe a rectangle $S$ in the triangle formed by the side of $R$ opposite the side on the boundary of $T,$ and the other two sides of $T,$ with one side along the side of $R.$ For any polygon $X,$ let $A(X)$ denote the area of $X.$ Find the maximum value, or show that no maximum exists, of $\tfrac{A(R)+A(S)}{A(T)},$ where $T$ ranges over all triangles and $R,S$ over all rectangles as above.
Find the value of $c$ such that the system of equations \begin{align*}|x+y|&=2007,\\|x-y|&=c\end{align*} has exactly two solutions $(x,y)$ in real numbers.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }223&\textbf{(U) }678\\\\
\textbf{(V) }2007 & &\end{array}$